TSTP Solution File: REL043+1 by Twee---2.4.2
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- Process Solution
%------------------------------------------------------------------------------
% File : Twee---2.4.2
% Problem : REL043+1 : TPTP v8.1.2. Released v4.0.0.
% Transfm : none
% Format : tptp:raw
% Command : parallel-twee %s --tstp --conditional-encoding if --smaller --drop-non-horn --give-up-on-saturation --explain-encoding --formal-proof
% Computer : n009.cluster.edu
% Model : x86_64 x86_64
% CPU : Intel(R) Xeon(R) CPU E5-2620 v4 2.10GHz
% Memory : 8042.1875MB
% OS : Linux 3.10.0-693.el7.x86_64
% CPULimit : 300s
% WCLimit : 300s
% DateTime : Thu Aug 31 13:44:28 EDT 2023
% Result : Theorem 35.68s 4.97s
% Output : Proof 36.35s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.07/0.12 % Problem : REL043+1 : TPTP v8.1.2. Released v4.0.0.
% 0.07/0.13 % Command : parallel-twee %s --tstp --conditional-encoding if --smaller --drop-non-horn --give-up-on-saturation --explain-encoding --formal-proof
% 0.13/0.34 % Computer : n009.cluster.edu
% 0.13/0.34 % Model : x86_64 x86_64
% 0.13/0.34 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.13/0.34 % Memory : 8042.1875MB
% 0.13/0.34 % OS : Linux 3.10.0-693.el7.x86_64
% 0.13/0.34 % CPULimit : 300
% 0.13/0.34 % WCLimit : 300
% 0.13/0.34 % DateTime : Fri Aug 25 22:04:20 EDT 2023
% 0.13/0.34 % CPUTime :
% 35.68/4.97 Command-line arguments: --flatten
% 35.68/4.97
% 35.68/4.97 % SZS status Theorem
% 35.68/4.97
% 36.35/5.02 % SZS output start Proof
% 36.35/5.02 Axiom 1 (converse_idempotence): converse(converse(X)) = X.
% 36.35/5.02 Axiom 2 (composition_identity): composition(X, one) = X.
% 36.35/5.02 Axiom 3 (maddux1_join_commutativity): join(X, Y) = join(Y, X).
% 36.35/5.02 Axiom 4 (def_zero): zero = meet(X, complement(X)).
% 36.35/5.02 Axiom 5 (def_top): top = join(X, complement(X)).
% 36.35/5.02 Axiom 6 (converse_multiplicativity): converse(composition(X, Y)) = composition(converse(Y), converse(X)).
% 36.35/5.02 Axiom 7 (composition_associativity): composition(X, composition(Y, Z)) = composition(composition(X, Y), Z).
% 36.35/5.02 Axiom 8 (converse_additivity): converse(join(X, Y)) = join(converse(X), converse(Y)).
% 36.35/5.02 Axiom 9 (maddux2_join_associativity): join(X, join(Y, Z)) = join(join(X, Y), Z).
% 36.35/5.02 Axiom 10 (goals): join(composition(x0, converse(x1)), x2) = x2.
% 36.35/5.02 Axiom 11 (maddux4_definiton_of_meet): meet(X, Y) = complement(join(complement(X), complement(Y))).
% 36.35/5.02 Axiom 12 (composition_distributivity): composition(join(X, Y), Z) = join(composition(X, Z), composition(Y, Z)).
% 36.35/5.02 Axiom 13 (converse_cancellativity): join(composition(converse(X), complement(composition(X, Y))), complement(Y)) = complement(Y).
% 36.35/5.02 Axiom 14 (maddux3_a_kind_of_de_Morgan): X = join(complement(join(complement(X), complement(Y))), complement(join(complement(X), Y))).
% 36.35/5.02
% 36.35/5.02 Lemma 15: complement(top) = zero.
% 36.35/5.02 Proof:
% 36.35/5.02 complement(top)
% 36.35/5.02 = { by axiom 5 (def_top) }
% 36.35/5.02 complement(join(complement(X), complement(complement(X))))
% 36.35/5.02 = { by axiom 11 (maddux4_definiton_of_meet) R->L }
% 36.35/5.02 meet(X, complement(X))
% 36.35/5.02 = { by axiom 4 (def_zero) R->L }
% 36.35/5.02 zero
% 36.35/5.02
% 36.35/5.02 Lemma 16: join(X, join(Y, complement(X))) = join(Y, top).
% 36.35/5.02 Proof:
% 36.35/5.02 join(X, join(Y, complement(X)))
% 36.35/5.02 = { by axiom 3 (maddux1_join_commutativity) R->L }
% 36.35/5.02 join(X, join(complement(X), Y))
% 36.35/5.02 = { by axiom 9 (maddux2_join_associativity) }
% 36.35/5.02 join(join(X, complement(X)), Y)
% 36.35/5.02 = { by axiom 5 (def_top) R->L }
% 36.35/5.02 join(top, Y)
% 36.35/5.02 = { by axiom 3 (maddux1_join_commutativity) }
% 36.35/5.02 join(Y, top)
% 36.35/5.02
% 36.35/5.02 Lemma 17: converse(composition(converse(X), Y)) = composition(converse(Y), X).
% 36.35/5.02 Proof:
% 36.35/5.02 converse(composition(converse(X), Y))
% 36.35/5.02 = { by axiom 6 (converse_multiplicativity) }
% 36.35/5.03 composition(converse(Y), converse(converse(X)))
% 36.35/5.03 = { by axiom 1 (converse_idempotence) }
% 36.35/5.03 composition(converse(Y), X)
% 36.35/5.03
% 36.35/5.03 Lemma 18: composition(converse(one), X) = X.
% 36.35/5.03 Proof:
% 36.35/5.03 composition(converse(one), X)
% 36.35/5.03 = { by lemma 17 R->L }
% 36.35/5.03 converse(composition(converse(X), one))
% 36.35/5.03 = { by axiom 2 (composition_identity) }
% 36.35/5.03 converse(converse(X))
% 36.35/5.03 = { by axiom 1 (converse_idempotence) }
% 36.35/5.03 X
% 36.35/5.03
% 36.35/5.03 Lemma 19: join(complement(X), composition(converse(Y), complement(composition(Y, X)))) = complement(X).
% 36.35/5.03 Proof:
% 36.35/5.03 join(complement(X), composition(converse(Y), complement(composition(Y, X))))
% 36.35/5.03 = { by axiom 3 (maddux1_join_commutativity) R->L }
% 36.35/5.03 join(composition(converse(Y), complement(composition(Y, X))), complement(X))
% 36.35/5.03 = { by axiom 13 (converse_cancellativity) }
% 36.35/5.03 complement(X)
% 36.35/5.03
% 36.35/5.03 Lemma 20: join(complement(X), complement(X)) = complement(X).
% 36.35/5.03 Proof:
% 36.35/5.03 join(complement(X), complement(X))
% 36.35/5.03 = { by lemma 18 R->L }
% 36.35/5.03 join(complement(X), composition(converse(one), complement(X)))
% 36.35/5.03 = { by lemma 18 R->L }
% 36.35/5.03 join(complement(X), composition(converse(one), complement(composition(converse(one), X))))
% 36.35/5.03 = { by axiom 2 (composition_identity) R->L }
% 36.35/5.03 join(complement(X), composition(converse(one), complement(composition(composition(converse(one), one), X))))
% 36.35/5.03 = { by axiom 7 (composition_associativity) R->L }
% 36.35/5.03 join(complement(X), composition(converse(one), complement(composition(converse(one), composition(one, X)))))
% 36.35/5.03 = { by lemma 18 }
% 36.35/5.03 join(complement(X), composition(converse(one), complement(composition(one, X))))
% 36.35/5.03 = { by lemma 19 }
% 36.35/5.03 complement(X)
% 36.35/5.03
% 36.35/5.03 Lemma 21: join(top, complement(X)) = top.
% 36.35/5.03 Proof:
% 36.35/5.03 join(top, complement(X))
% 36.35/5.03 = { by axiom 3 (maddux1_join_commutativity) R->L }
% 36.35/5.03 join(complement(X), top)
% 36.35/5.03 = { by lemma 16 R->L }
% 36.35/5.03 join(X, join(complement(X), complement(X)))
% 36.35/5.03 = { by lemma 20 }
% 36.35/5.03 join(X, complement(X))
% 36.35/5.03 = { by axiom 5 (def_top) R->L }
% 36.35/5.03 top
% 36.35/5.03
% 36.35/5.03 Lemma 22: join(Y, top) = join(X, top).
% 36.35/5.03 Proof:
% 36.35/5.03 join(Y, top)
% 36.35/5.03 = { by lemma 21 R->L }
% 36.35/5.03 join(Y, join(top, complement(Y)))
% 36.35/5.03 = { by lemma 16 }
% 36.35/5.03 join(top, top)
% 36.35/5.03 = { by lemma 16 R->L }
% 36.35/5.03 join(X, join(top, complement(X)))
% 36.35/5.03 = { by lemma 21 }
% 36.35/5.03 join(X, top)
% 36.35/5.03
% 36.35/5.03 Lemma 23: join(X, top) = top.
% 36.35/5.03 Proof:
% 36.35/5.03 join(X, top)
% 36.35/5.03 = { by lemma 22 }
% 36.35/5.03 join(complement(Y), top)
% 36.35/5.03 = { by axiom 3 (maddux1_join_commutativity) R->L }
% 36.35/5.03 join(top, complement(Y))
% 36.35/5.03 = { by lemma 21 }
% 36.35/5.03 top
% 36.35/5.03
% 36.35/5.03 Lemma 24: join(meet(X, Y), complement(join(complement(X), Y))) = X.
% 36.35/5.03 Proof:
% 36.35/5.03 join(meet(X, Y), complement(join(complement(X), Y)))
% 36.35/5.03 = { by axiom 11 (maddux4_definiton_of_meet) }
% 36.35/5.03 join(complement(join(complement(X), complement(Y))), complement(join(complement(X), Y)))
% 36.35/5.03 = { by axiom 14 (maddux3_a_kind_of_de_Morgan) R->L }
% 36.35/5.03 X
% 36.35/5.03
% 36.35/5.03 Lemma 25: join(zero, meet(X, X)) = X.
% 36.35/5.03 Proof:
% 36.35/5.03 join(zero, meet(X, X))
% 36.35/5.03 = { by axiom 11 (maddux4_definiton_of_meet) }
% 36.35/5.03 join(zero, complement(join(complement(X), complement(X))))
% 36.35/5.03 = { by axiom 4 (def_zero) }
% 36.35/5.03 join(meet(X, complement(X)), complement(join(complement(X), complement(X))))
% 36.35/5.03 = { by lemma 24 }
% 36.35/5.03 X
% 36.35/5.03
% 36.35/5.03 Lemma 26: complement(complement(X)) = meet(X, X).
% 36.35/5.03 Proof:
% 36.35/5.03 complement(complement(X))
% 36.35/5.03 = { by lemma 20 R->L }
% 36.35/5.03 complement(join(complement(X), complement(X)))
% 36.35/5.03 = { by axiom 11 (maddux4_definiton_of_meet) R->L }
% 36.35/5.03 meet(X, X)
% 36.35/5.03
% 36.35/5.03 Lemma 27: meet(Y, X) = meet(X, Y).
% 36.35/5.03 Proof:
% 36.35/5.03 meet(Y, X)
% 36.35/5.03 = { by axiom 11 (maddux4_definiton_of_meet) }
% 36.35/5.03 complement(join(complement(Y), complement(X)))
% 36.35/5.03 = { by axiom 3 (maddux1_join_commutativity) R->L }
% 36.35/5.03 complement(join(complement(X), complement(Y)))
% 36.35/5.03 = { by axiom 11 (maddux4_definiton_of_meet) R->L }
% 36.35/5.03 meet(X, Y)
% 36.35/5.03
% 36.35/5.03 Lemma 28: complement(join(zero, complement(X))) = meet(X, top).
% 36.35/5.03 Proof:
% 36.35/5.03 complement(join(zero, complement(X)))
% 36.35/5.03 = { by lemma 15 R->L }
% 36.35/5.03 complement(join(complement(top), complement(X)))
% 36.35/5.03 = { by axiom 11 (maddux4_definiton_of_meet) R->L }
% 36.35/5.03 meet(top, X)
% 36.35/5.03 = { by lemma 27 R->L }
% 36.35/5.03 meet(X, top)
% 36.35/5.03
% 36.35/5.03 Lemma 29: join(X, join(complement(X), Y)) = top.
% 36.35/5.03 Proof:
% 36.35/5.03 join(X, join(complement(X), Y))
% 36.35/5.03 = { by axiom 3 (maddux1_join_commutativity) R->L }
% 36.35/5.03 join(X, join(Y, complement(X)))
% 36.35/5.03 = { by lemma 16 }
% 36.35/5.03 join(Y, top)
% 36.35/5.03 = { by lemma 22 R->L }
% 36.35/5.03 join(Z, top)
% 36.35/5.03 = { by lemma 23 }
% 36.35/5.03 top
% 36.35/5.03
% 36.35/5.03 Lemma 30: join(X, complement(zero)) = top.
% 36.35/5.03 Proof:
% 36.35/5.03 join(X, complement(zero))
% 36.35/5.03 = { by lemma 25 R->L }
% 36.35/5.03 join(join(zero, meet(X, X)), complement(zero))
% 36.35/5.03 = { by axiom 9 (maddux2_join_associativity) R->L }
% 36.35/5.03 join(zero, join(meet(X, X), complement(zero)))
% 36.35/5.03 = { by axiom 3 (maddux1_join_commutativity) }
% 36.35/5.03 join(zero, join(complement(zero), meet(X, X)))
% 36.35/5.03 = { by lemma 29 }
% 36.35/5.03 top
% 36.35/5.03
% 36.35/5.03 Lemma 31: join(meet(X, Y), meet(X, complement(Y))) = X.
% 36.35/5.03 Proof:
% 36.35/5.03 join(meet(X, Y), meet(X, complement(Y)))
% 36.35/5.03 = { by axiom 3 (maddux1_join_commutativity) R->L }
% 36.35/5.03 join(meet(X, complement(Y)), meet(X, Y))
% 36.35/5.03 = { by axiom 11 (maddux4_definiton_of_meet) }
% 36.35/5.03 join(meet(X, complement(Y)), complement(join(complement(X), complement(Y))))
% 36.35/5.03 = { by lemma 24 }
% 36.35/5.03 X
% 36.35/5.03
% 36.35/5.03 Lemma 32: join(zero, meet(X, top)) = X.
% 36.35/5.03 Proof:
% 36.35/5.03 join(zero, meet(X, top))
% 36.35/5.03 = { by lemma 30 R->L }
% 36.35/5.03 join(zero, meet(X, join(complement(zero), complement(zero))))
% 36.35/5.03 = { by lemma 20 }
% 36.35/5.03 join(zero, meet(X, complement(zero)))
% 36.35/5.03 = { by lemma 15 R->L }
% 36.35/5.03 join(complement(top), meet(X, complement(zero)))
% 36.35/5.03 = { by lemma 30 R->L }
% 36.35/5.03 join(complement(join(complement(X), complement(zero))), meet(X, complement(zero)))
% 36.35/5.03 = { by axiom 11 (maddux4_definiton_of_meet) R->L }
% 36.35/5.03 join(meet(X, zero), meet(X, complement(zero)))
% 36.35/5.03 = { by lemma 31 }
% 36.35/5.03 X
% 36.35/5.03
% 36.35/5.03 Lemma 33: join(zero, complement(X)) = complement(X).
% 36.35/5.03 Proof:
% 36.35/5.03 join(zero, complement(X))
% 36.35/5.03 = { by lemma 25 R->L }
% 36.35/5.03 join(zero, complement(join(zero, meet(X, X))))
% 36.35/5.03 = { by lemma 26 R->L }
% 36.35/5.03 join(zero, complement(join(zero, complement(complement(X)))))
% 36.35/5.03 = { by lemma 28 }
% 36.35/5.03 join(zero, meet(complement(X), top))
% 36.35/5.03 = { by lemma 32 }
% 36.35/5.03 complement(X)
% 36.35/5.03
% 36.35/5.03 Lemma 34: complement(complement(X)) = X.
% 36.35/5.03 Proof:
% 36.35/5.03 complement(complement(X))
% 36.35/5.03 = { by lemma 33 R->L }
% 36.35/5.03 join(zero, complement(complement(X)))
% 36.35/5.03 = { by lemma 26 }
% 36.35/5.03 join(zero, meet(X, X))
% 36.35/5.03 = { by lemma 25 }
% 36.35/5.03 X
% 36.35/5.03
% 36.35/5.03 Lemma 35: join(X, zero) = X.
% 36.35/5.03 Proof:
% 36.35/5.03 join(X, zero)
% 36.35/5.03 = { by axiom 3 (maddux1_join_commutativity) R->L }
% 36.35/5.03 join(zero, X)
% 36.35/5.03 = { by lemma 34 R->L }
% 36.35/5.03 join(zero, complement(complement(X)))
% 36.35/5.03 = { by lemma 26 }
% 36.35/5.03 join(zero, meet(X, X))
% 36.35/5.03 = { by lemma 25 }
% 36.35/5.03 X
% 36.35/5.03
% 36.35/5.03 Lemma 36: join(zero, X) = X.
% 36.35/5.03 Proof:
% 36.35/5.03 join(zero, X)
% 36.35/5.03 = { by axiom 3 (maddux1_join_commutativity) R->L }
% 36.35/5.03 join(X, zero)
% 36.35/5.03 = { by lemma 35 }
% 36.35/5.03 X
% 36.35/5.03
% 36.35/5.03 Lemma 37: complement(join(complement(X), meet(Y, Z))) = meet(X, join(complement(Y), complement(Z))).
% 36.35/5.03 Proof:
% 36.35/5.03 complement(join(complement(X), meet(Y, Z)))
% 36.35/5.03 = { by lemma 27 }
% 36.35/5.03 complement(join(complement(X), meet(Z, Y)))
% 36.35/5.03 = { by axiom 3 (maddux1_join_commutativity) R->L }
% 36.35/5.03 complement(join(meet(Z, Y), complement(X)))
% 36.35/5.03 = { by axiom 11 (maddux4_definiton_of_meet) }
% 36.35/5.03 complement(join(complement(join(complement(Z), complement(Y))), complement(X)))
% 36.35/5.03 = { by axiom 11 (maddux4_definiton_of_meet) R->L }
% 36.35/5.03 meet(join(complement(Z), complement(Y)), X)
% 36.35/5.03 = { by lemma 27 R->L }
% 36.35/5.03 meet(X, join(complement(Z), complement(Y)))
% 36.35/5.03 = { by axiom 3 (maddux1_join_commutativity) }
% 36.35/5.03 meet(X, join(complement(Y), complement(Z)))
% 36.35/5.03
% 36.35/5.03 Lemma 38: meet(X, top) = X.
% 36.35/5.03 Proof:
% 36.35/5.03 meet(X, top)
% 36.35/5.03 = { by lemma 28 R->L }
% 36.35/5.03 complement(join(zero, complement(X)))
% 36.35/5.03 = { by lemma 33 R->L }
% 36.35/5.03 join(zero, complement(join(zero, complement(X))))
% 36.35/5.03 = { by lemma 28 }
% 36.35/5.03 join(zero, meet(X, top))
% 36.35/5.03 = { by lemma 32 }
% 36.35/5.03 X
% 36.35/5.03
% 36.35/5.03 Lemma 39: meet(top, X) = X.
% 36.35/5.03 Proof:
% 36.35/5.03 meet(top, X)
% 36.35/5.03 = { by lemma 27 }
% 36.35/5.03 meet(X, top)
% 36.35/5.03 = { by lemma 38 }
% 36.35/5.03 X
% 36.35/5.03
% 36.35/5.03 Lemma 40: complement(join(zero, meet(X, Y))) = join(complement(X), complement(Y)).
% 36.35/5.03 Proof:
% 36.35/5.03 complement(join(zero, meet(X, Y)))
% 36.35/5.03 = { by lemma 27 }
% 36.35/5.03 complement(join(zero, meet(Y, X)))
% 36.35/5.03 = { by lemma 15 R->L }
% 36.35/5.03 complement(join(complement(top), meet(Y, X)))
% 36.35/5.03 = { by lemma 37 }
% 36.35/5.03 meet(top, join(complement(Y), complement(X)))
% 36.35/5.03 = { by lemma 39 }
% 36.35/5.03 join(complement(Y), complement(X))
% 36.35/5.03 = { by axiom 3 (maddux1_join_commutativity) }
% 36.35/5.03 join(complement(X), complement(Y))
% 36.35/5.03
% 36.35/5.03 Lemma 41: join(complement(X), complement(Y)) = complement(meet(X, Y)).
% 36.35/5.03 Proof:
% 36.35/5.03 join(complement(X), complement(Y))
% 36.35/5.03 = { by axiom 3 (maddux1_join_commutativity) R->L }
% 36.35/5.03 join(complement(Y), complement(X))
% 36.35/5.03 = { by lemma 40 R->L }
% 36.35/5.03 complement(join(zero, meet(Y, X)))
% 36.35/5.03 = { by axiom 3 (maddux1_join_commutativity) }
% 36.35/5.03 complement(join(meet(Y, X), zero))
% 36.35/5.03 = { by lemma 35 }
% 36.35/5.03 complement(meet(Y, X))
% 36.35/5.03 = { by lemma 27 R->L }
% 36.35/5.03 complement(meet(X, Y))
% 36.35/5.03
% 36.35/5.03 Lemma 42: complement(meet(X, complement(Y))) = join(Y, complement(X)).
% 36.35/5.03 Proof:
% 36.35/5.03 complement(meet(X, complement(Y)))
% 36.35/5.03 = { by lemma 27 }
% 36.35/5.03 complement(meet(complement(Y), X))
% 36.35/5.03 = { by lemma 33 R->L }
% 36.35/5.03 complement(meet(join(zero, complement(Y)), X))
% 36.35/5.03 = { by lemma 41 R->L }
% 36.35/5.03 join(complement(join(zero, complement(Y))), complement(X))
% 36.35/5.03 = { by lemma 28 }
% 36.35/5.03 join(meet(Y, top), complement(X))
% 36.35/5.03 = { by lemma 38 }
% 36.35/5.03 join(Y, complement(X))
% 36.35/5.03
% 36.35/5.03 Lemma 43: meet(X, X) = X.
% 36.35/5.03 Proof:
% 36.35/5.03 meet(X, X)
% 36.35/5.03 = { by lemma 26 R->L }
% 36.35/5.03 complement(complement(X))
% 36.35/5.03 = { by lemma 34 }
% 36.35/5.03 X
% 36.35/5.03
% 36.35/5.03 Lemma 44: meet(meet(X, Y), Z) = meet(X, meet(Z, Y)).
% 36.35/5.03 Proof:
% 36.35/5.03 meet(meet(X, Y), Z)
% 36.35/5.03 = { by lemma 27 }
% 36.35/5.03 meet(Z, meet(X, Y))
% 36.35/5.03 = { by lemma 43 R->L }
% 36.35/5.03 meet(meet(Z, meet(X, Y)), meet(Z, meet(X, Y)))
% 36.35/5.03 = { by lemma 26 R->L }
% 36.35/5.03 complement(complement(meet(Z, meet(X, Y))))
% 36.35/5.03 = { by lemma 27 }
% 36.35/5.03 complement(complement(meet(Z, meet(Y, X))))
% 36.35/5.03 = { by lemma 41 R->L }
% 36.35/5.03 complement(join(complement(Z), complement(meet(Y, X))))
% 36.35/5.03 = { by lemma 41 R->L }
% 36.35/5.03 complement(join(complement(Z), join(complement(Y), complement(X))))
% 36.35/5.03 = { by axiom 9 (maddux2_join_associativity) }
% 36.35/5.03 complement(join(join(complement(Z), complement(Y)), complement(X)))
% 36.35/5.03 = { by lemma 42 R->L }
% 36.35/5.03 complement(complement(meet(X, complement(join(complement(Z), complement(Y))))))
% 36.35/5.03 = { by axiom 11 (maddux4_definiton_of_meet) R->L }
% 36.35/5.03 complement(complement(meet(X, meet(Z, Y))))
% 36.35/5.03 = { by lemma 27 R->L }
% 36.35/5.03 complement(complement(meet(X, meet(Y, Z))))
% 36.35/5.03 = { by lemma 34 }
% 36.35/5.03 meet(X, meet(Y, Z))
% 36.35/5.03 = { by lemma 27 R->L }
% 36.35/5.03 meet(X, meet(Z, Y))
% 36.35/5.03
% 36.35/5.03 Lemma 45: meet(X, meet(complement(Y), join(Y, complement(X)))) = zero.
% 36.35/5.03 Proof:
% 36.35/5.03 meet(X, meet(complement(Y), join(Y, complement(X))))
% 36.35/5.03 = { by lemma 27 }
% 36.35/5.03 meet(X, meet(join(Y, complement(X)), complement(Y)))
% 36.35/5.03 = { by lemma 44 R->L }
% 36.35/5.03 meet(meet(X, complement(Y)), join(Y, complement(X)))
% 36.35/5.03 = { by lemma 42 R->L }
% 36.35/5.03 meet(meet(X, complement(Y)), complement(meet(X, complement(Y))))
% 36.35/5.03 = { by axiom 4 (def_zero) R->L }
% 36.35/5.03 zero
% 36.35/5.03
% 36.35/5.03 Lemma 46: converse(join(X, converse(Y))) = join(Y, converse(X)).
% 36.35/5.03 Proof:
% 36.35/5.03 converse(join(X, converse(Y)))
% 36.35/5.03 = { by axiom 3 (maddux1_join_commutativity) R->L }
% 36.35/5.03 converse(join(converse(Y), X))
% 36.35/5.03 = { by axiom 8 (converse_additivity) }
% 36.35/5.03 join(converse(converse(Y)), converse(X))
% 36.35/5.03 = { by axiom 1 (converse_idempotence) }
% 36.35/5.03 join(Y, converse(X))
% 36.35/5.03
% 36.35/5.03 Lemma 47: converse(join(converse(X), Y)) = join(X, converse(Y)).
% 36.35/5.03 Proof:
% 36.35/5.03 converse(join(converse(X), Y))
% 36.35/5.03 = { by axiom 3 (maddux1_join_commutativity) R->L }
% 36.35/5.03 converse(join(Y, converse(X)))
% 36.35/5.03 = { by lemma 46 }
% 36.35/5.03 join(X, converse(Y))
% 36.35/5.03
% 36.35/5.03 Lemma 48: meet(complement(X), converse(complement(converse(X)))) = complement(X).
% 36.35/5.03 Proof:
% 36.35/5.03 meet(complement(X), converse(complement(converse(X))))
% 36.35/5.03 = { by lemma 24 R->L }
% 36.35/5.03 meet(complement(X), join(meet(converse(complement(converse(X))), meet(complement(X), complement(converse(complement(converse(X)))))), complement(join(complement(converse(complement(converse(X)))), meet(complement(X), complement(converse(complement(converse(X)))))))))
% 36.35/5.03 = { by lemma 27 }
% 36.35/5.03 meet(complement(X), join(meet(converse(complement(converse(X))), meet(complement(converse(complement(converse(X)))), complement(X))), complement(join(complement(converse(complement(converse(X)))), meet(complement(X), complement(converse(complement(converse(X)))))))))
% 36.35/5.03 = { by axiom 11 (maddux4_definiton_of_meet) }
% 36.35/5.03 meet(complement(X), join(complement(join(complement(converse(complement(converse(X)))), complement(meet(complement(converse(complement(converse(X)))), complement(X))))), complement(join(complement(converse(complement(converse(X)))), meet(complement(X), complement(converse(complement(converse(X)))))))))
% 36.35/5.04 = { by lemma 27 }
% 36.35/5.04 meet(complement(X), join(complement(join(complement(converse(complement(converse(X)))), complement(meet(complement(X), complement(converse(complement(converse(X)))))))), complement(join(complement(converse(complement(converse(X)))), meet(complement(X), complement(converse(complement(converse(X)))))))))
% 36.35/5.04 = { by lemma 41 R->L }
% 36.35/5.04 meet(complement(X), join(complement(join(complement(converse(complement(converse(X)))), join(complement(complement(X)), complement(complement(converse(complement(converse(X)))))))), complement(join(complement(converse(complement(converse(X)))), meet(complement(X), complement(converse(complement(converse(X)))))))))
% 36.35/5.04 = { by lemma 16 }
% 36.35/5.04 meet(complement(X), join(complement(join(complement(complement(X)), top)), complement(join(complement(converse(complement(converse(X)))), meet(complement(X), complement(converse(complement(converse(X)))))))))
% 36.35/5.04 = { by lemma 23 }
% 36.35/5.04 meet(complement(X), join(complement(top), complement(join(complement(converse(complement(converse(X)))), meet(complement(X), complement(converse(complement(converse(X)))))))))
% 36.35/5.04 = { by lemma 15 }
% 36.35/5.04 meet(complement(X), join(zero, complement(join(complement(converse(complement(converse(X)))), meet(complement(X), complement(converse(complement(converse(X)))))))))
% 36.35/5.04 = { by lemma 33 }
% 36.35/5.04 meet(complement(X), complement(join(complement(converse(complement(converse(X)))), meet(complement(X), complement(converse(complement(converse(X))))))))
% 36.35/5.04 = { by lemma 37 }
% 36.35/5.04 meet(complement(X), meet(converse(complement(converse(X))), join(complement(complement(X)), complement(complement(converse(complement(converse(X))))))))
% 36.35/5.04 = { by lemma 41 }
% 36.35/5.04 meet(complement(X), meet(converse(complement(converse(X))), complement(meet(complement(X), complement(converse(complement(converse(X))))))))
% 36.35/5.04 = { by lemma 42 }
% 36.35/5.04 meet(complement(X), meet(converse(complement(converse(X))), join(converse(complement(converse(X))), complement(complement(X)))))
% 36.35/5.04 = { by lemma 26 }
% 36.35/5.04 meet(complement(X), meet(converse(complement(converse(X))), join(converse(complement(converse(X))), meet(X, X))))
% 36.35/5.04 = { by lemma 43 }
% 36.35/5.04 meet(complement(X), meet(converse(complement(converse(X))), join(converse(complement(converse(X))), X)))
% 36.35/5.04 = { by axiom 3 (maddux1_join_commutativity) }
% 36.35/5.04 meet(complement(X), meet(converse(complement(converse(X))), join(X, converse(complement(converse(X))))))
% 36.35/5.04 = { by lemma 34 R->L }
% 36.35/5.04 meet(complement(X), meet(converse(complement(converse(X))), join(X, complement(complement(converse(complement(converse(X))))))))
% 36.35/5.04 = { by lemma 35 R->L }
% 36.35/5.04 meet(complement(X), meet(join(converse(complement(converse(X))), zero), join(X, complement(complement(converse(complement(converse(X))))))))
% 36.35/5.04 = { by lemma 44 R->L }
% 36.35/5.04 meet(meet(complement(X), join(X, complement(complement(converse(complement(converse(X))))))), join(converse(complement(converse(X))), zero))
% 36.35/5.04 = { by lemma 45 R->L }
% 36.35/5.04 meet(meet(complement(X), join(X, complement(complement(converse(complement(converse(X))))))), join(converse(complement(converse(X))), meet(complement(converse(complement(converse(X)))), meet(complement(X), join(X, complement(complement(converse(complement(converse(X))))))))))
% 36.35/5.04 = { by lemma 27 }
% 36.35/5.04 meet(meet(complement(X), join(X, complement(complement(converse(complement(converse(X))))))), join(converse(complement(converse(X))), meet(meet(complement(X), join(X, complement(complement(converse(complement(converse(X))))))), complement(converse(complement(converse(X)))))))
% 36.35/5.04 = { by lemma 33 R->L }
% 36.35/5.04 meet(meet(complement(X), join(X, complement(complement(converse(complement(converse(X))))))), join(converse(complement(converse(X))), meet(meet(complement(X), join(X, complement(complement(converse(complement(converse(X))))))), join(zero, complement(converse(complement(converse(X))))))))
% 36.35/5.04 = { by lemma 38 R->L }
% 36.35/5.04 meet(meet(complement(X), join(X, complement(complement(converse(complement(converse(X))))))), join(meet(converse(complement(converse(X))), top), meet(meet(complement(X), join(X, complement(complement(converse(complement(converse(X))))))), join(zero, complement(converse(complement(converse(X))))))))
% 36.35/5.04 = { by lemma 28 R->L }
% 36.35/5.04 meet(meet(complement(X), join(X, complement(complement(converse(complement(converse(X))))))), join(complement(join(zero, complement(converse(complement(converse(X)))))), meet(meet(complement(X), join(X, complement(complement(converse(complement(converse(X))))))), join(zero, complement(converse(complement(converse(X))))))))
% 36.35/5.04 = { by lemma 35 R->L }
% 36.35/5.04 join(meet(meet(complement(X), join(X, complement(complement(converse(complement(converse(X))))))), join(complement(join(zero, complement(converse(complement(converse(X)))))), meet(meet(complement(X), join(X, complement(complement(converse(complement(converse(X))))))), join(zero, complement(converse(complement(converse(X)))))))), zero)
% 36.35/5.04 = { by lemma 15 R->L }
% 36.35/5.04 join(meet(meet(complement(X), join(X, complement(complement(converse(complement(converse(X))))))), join(complement(join(zero, complement(converse(complement(converse(X)))))), meet(meet(complement(X), join(X, complement(complement(converse(complement(converse(X))))))), join(zero, complement(converse(complement(converse(X)))))))), complement(top))
% 36.35/5.04 = { by axiom 5 (def_top) }
% 36.35/5.04 join(meet(meet(complement(X), join(X, complement(complement(converse(complement(converse(X))))))), join(complement(join(zero, complement(converse(complement(converse(X)))))), meet(meet(complement(X), join(X, complement(complement(converse(complement(converse(X))))))), join(zero, complement(converse(complement(converse(X)))))))), complement(join(join(complement(meet(complement(X), join(X, complement(complement(converse(complement(converse(X)))))))), complement(join(zero, complement(converse(complement(converse(X))))))), complement(join(complement(meet(complement(X), join(X, complement(complement(converse(complement(converse(X)))))))), complement(join(zero, complement(converse(complement(converse(X)))))))))))
% 36.35/5.04 = { by axiom 11 (maddux4_definiton_of_meet) R->L }
% 36.35/5.04 join(meet(meet(complement(X), join(X, complement(complement(converse(complement(converse(X))))))), join(complement(join(zero, complement(converse(complement(converse(X)))))), meet(meet(complement(X), join(X, complement(complement(converse(complement(converse(X))))))), join(zero, complement(converse(complement(converse(X)))))))), complement(join(join(complement(meet(complement(X), join(X, complement(complement(converse(complement(converse(X)))))))), complement(join(zero, complement(converse(complement(converse(X))))))), meet(meet(complement(X), join(X, complement(complement(converse(complement(converse(X))))))), join(zero, complement(converse(complement(converse(X)))))))))
% 36.35/5.04 = { by axiom 9 (maddux2_join_associativity) R->L }
% 36.35/5.04 join(meet(meet(complement(X), join(X, complement(complement(converse(complement(converse(X))))))), join(complement(join(zero, complement(converse(complement(converse(X)))))), meet(meet(complement(X), join(X, complement(complement(converse(complement(converse(X))))))), join(zero, complement(converse(complement(converse(X)))))))), complement(join(complement(meet(complement(X), join(X, complement(complement(converse(complement(converse(X)))))))), join(complement(join(zero, complement(converse(complement(converse(X)))))), meet(meet(complement(X), join(X, complement(complement(converse(complement(converse(X))))))), join(zero, complement(converse(complement(converse(X))))))))))
% 36.35/5.04 = { by lemma 24 }
% 36.35/5.04 meet(complement(X), join(X, complement(complement(converse(complement(converse(X)))))))
% 36.35/5.04 = { by lemma 34 }
% 36.35/5.04 meet(complement(X), join(X, converse(complement(converse(X)))))
% 36.35/5.04 = { by lemma 47 R->L }
% 36.35/5.04 meet(complement(X), converse(join(converse(X), complement(converse(X)))))
% 36.35/5.04 = { by axiom 5 (def_top) R->L }
% 36.35/5.04 meet(complement(X), converse(top))
% 36.35/5.04 = { by lemma 23 R->L }
% 36.35/5.04 meet(complement(X), converse(join(Y, top)))
% 36.35/5.04 = { by axiom 8 (converse_additivity) }
% 36.35/5.04 meet(complement(X), join(converse(Y), converse(top)))
% 36.35/5.04 = { by axiom 5 (def_top) }
% 36.35/5.04 meet(complement(X), join(converse(Y), converse(join(converse(complement(converse(Y))), complement(converse(complement(converse(Y))))))))
% 36.35/5.04 = { by lemma 47 }
% 36.35/5.04 meet(complement(X), join(converse(Y), join(complement(converse(Y)), converse(complement(converse(complement(converse(Y))))))))
% 36.35/5.04 = { by lemma 29 }
% 36.35/5.04 meet(complement(X), top)
% 36.35/5.04 = { by lemma 38 }
% 36.35/5.04 complement(X)
% 36.35/5.04
% 36.35/5.04 Lemma 49: complement(converse(X)) = converse(complement(X)).
% 36.35/5.04 Proof:
% 36.35/5.04 complement(converse(X))
% 36.35/5.04 = { by axiom 1 (converse_idempotence) R->L }
% 36.35/5.04 converse(converse(complement(converse(X))))
% 36.35/5.04 = { by lemma 31 R->L }
% 36.35/5.04 converse(join(meet(converse(complement(converse(X))), X), meet(converse(complement(converse(X))), complement(X))))
% 36.35/5.04 = { by lemma 27 R->L }
% 36.35/5.04 converse(join(meet(X, converse(complement(converse(X)))), meet(converse(complement(converse(X))), complement(X))))
% 36.35/5.04 = { by lemma 27 R->L }
% 36.35/5.04 converse(join(meet(X, converse(complement(converse(X)))), meet(complement(X), converse(complement(converse(X))))))
% 36.35/5.04 = { by axiom 1 (converse_idempotence) R->L }
% 36.35/5.04 converse(join(meet(converse(converse(X)), converse(complement(converse(X)))), meet(complement(X), converse(complement(converse(X))))))
% 36.35/5.04 = { by lemma 34 R->L }
% 36.35/5.04 converse(join(meet(complement(complement(converse(converse(X)))), converse(complement(converse(X)))), meet(complement(X), converse(complement(converse(X))))))
% 36.35/5.04 = { by lemma 48 R->L }
% 36.35/5.04 converse(join(meet(complement(complement(converse(converse(X)))), converse(meet(complement(converse(X)), converse(complement(converse(converse(X))))))), meet(complement(X), converse(complement(converse(X))))))
% 36.35/5.04 = { by lemma 27 }
% 36.35/5.04 converse(join(meet(converse(meet(complement(converse(X)), converse(complement(converse(converse(X)))))), complement(complement(converse(converse(X))))), meet(complement(X), converse(complement(converse(X))))))
% 36.35/5.04 = { by lemma 38 R->L }
% 36.35/5.04 converse(join(meet(converse(meet(complement(converse(X)), converse(complement(converse(converse(X)))))), meet(complement(complement(converse(converse(X)))), top)), meet(complement(X), converse(complement(converse(X))))))
% 36.35/5.04 = { by lemma 29 R->L }
% 36.35/5.04 converse(join(meet(converse(meet(complement(converse(X)), converse(complement(converse(converse(X)))))), meet(complement(complement(converse(converse(X)))), join(converse(meet(converse(complement(converse(converse(X)))), complement(converse(X)))), join(complement(converse(meet(converse(complement(converse(converse(X)))), complement(converse(X))))), converse(complement(join(complement(converse(complement(converse(converse(X))))), complement(converse(X))))))))), meet(complement(X), converse(complement(converse(X))))))
% 36.35/5.04 = { by axiom 3 (maddux1_join_commutativity) R->L }
% 36.35/5.04 converse(join(meet(converse(meet(complement(converse(X)), converse(complement(converse(converse(X)))))), meet(complement(complement(converse(converse(X)))), join(converse(meet(converse(complement(converse(converse(X)))), complement(converse(X)))), join(converse(complement(join(complement(converse(complement(converse(converse(X))))), complement(converse(X))))), complement(converse(meet(converse(complement(converse(converse(X)))), complement(converse(X))))))))), meet(complement(X), converse(complement(converse(X))))))
% 36.35/5.04 = { by axiom 9 (maddux2_join_associativity) }
% 36.35/5.04 converse(join(meet(converse(meet(complement(converse(X)), converse(complement(converse(converse(X)))))), meet(complement(complement(converse(converse(X)))), join(join(converse(meet(converse(complement(converse(converse(X)))), complement(converse(X)))), converse(complement(join(complement(converse(complement(converse(converse(X))))), complement(converse(X)))))), complement(converse(meet(converse(complement(converse(converse(X)))), complement(converse(X)))))))), meet(complement(X), converse(complement(converse(X))))))
% 36.35/5.04 = { by axiom 8 (converse_additivity) R->L }
% 36.35/5.04 converse(join(meet(converse(meet(complement(converse(X)), converse(complement(converse(converse(X)))))), meet(complement(complement(converse(converse(X)))), join(converse(join(meet(converse(complement(converse(converse(X)))), complement(converse(X))), complement(join(complement(converse(complement(converse(converse(X))))), complement(converse(X)))))), complement(converse(meet(converse(complement(converse(converse(X)))), complement(converse(X)))))))), meet(complement(X), converse(complement(converse(X))))))
% 36.35/5.04 = { by axiom 3 (maddux1_join_commutativity) }
% 36.35/5.04 converse(join(meet(converse(meet(complement(converse(X)), converse(complement(converse(converse(X)))))), meet(complement(complement(converse(converse(X)))), join(complement(converse(meet(converse(complement(converse(converse(X)))), complement(converse(X))))), converse(join(meet(converse(complement(converse(converse(X)))), complement(converse(X))), complement(join(complement(converse(complement(converse(converse(X))))), complement(converse(X))))))))), meet(complement(X), converse(complement(converse(X))))))
% 36.35/5.04 = { by lemma 24 }
% 36.35/5.04 converse(join(meet(converse(meet(complement(converse(X)), converse(complement(converse(converse(X)))))), meet(complement(complement(converse(converse(X)))), join(complement(converse(meet(converse(complement(converse(converse(X)))), complement(converse(X))))), converse(converse(complement(converse(converse(X)))))))), meet(complement(X), converse(complement(converse(X))))))
% 36.35/5.04 = { by axiom 3 (maddux1_join_commutativity) }
% 36.35/5.04 converse(join(meet(converse(meet(complement(converse(X)), converse(complement(converse(converse(X)))))), meet(complement(complement(converse(converse(X)))), join(converse(converse(complement(converse(converse(X))))), complement(converse(meet(converse(complement(converse(converse(X)))), complement(converse(X)))))))), meet(complement(X), converse(complement(converse(X))))))
% 36.35/5.04 = { by axiom 1 (converse_idempotence) }
% 36.35/5.04 converse(join(meet(converse(meet(complement(converse(X)), converse(complement(converse(converse(X)))))), meet(complement(complement(converse(converse(X)))), join(complement(converse(converse(X))), complement(converse(meet(converse(complement(converse(converse(X)))), complement(converse(X)))))))), meet(complement(X), converse(complement(converse(X))))))
% 36.35/5.04 = { by lemma 27 R->L }
% 36.35/5.04 converse(join(meet(converse(meet(complement(converse(X)), converse(complement(converse(converse(X)))))), meet(complement(complement(converse(converse(X)))), join(complement(converse(converse(X))), complement(converse(meet(complement(converse(X)), converse(complement(converse(converse(X)))))))))), meet(complement(X), converse(complement(converse(X))))))
% 36.35/5.04 = { by lemma 45 }
% 36.35/5.04 converse(join(zero, meet(complement(X), converse(complement(converse(X))))))
% 36.35/5.05 = { by lemma 36 }
% 36.35/5.05 converse(meet(complement(X), converse(complement(converse(X)))))
% 36.35/5.05 = { by lemma 48 }
% 36.35/5.05 converse(complement(X))
% 36.35/5.05
% 36.35/5.05 Lemma 50: join(complement(X), composition(complement(composition(X, converse(Y))), Y)) = complement(X).
% 36.35/5.05 Proof:
% 36.35/5.05 join(complement(X), composition(complement(composition(X, converse(Y))), Y))
% 36.35/5.05 = { by axiom 1 (converse_idempotence) R->L }
% 36.35/5.05 converse(converse(join(complement(X), composition(complement(composition(X, converse(Y))), Y))))
% 36.35/5.05 = { by axiom 1 (converse_idempotence) R->L }
% 36.35/5.05 converse(converse(join(complement(X), composition(complement(composition(X, converse(Y))), converse(converse(Y))))))
% 36.35/5.05 = { by axiom 3 (maddux1_join_commutativity) R->L }
% 36.35/5.05 converse(converse(join(composition(complement(composition(X, converse(Y))), converse(converse(Y))), complement(X))))
% 36.35/5.05 = { by axiom 8 (converse_additivity) }
% 36.35/5.05 converse(join(converse(composition(complement(composition(X, converse(Y))), converse(converse(Y)))), converse(complement(X))))
% 36.35/5.05 = { by axiom 6 (converse_multiplicativity) }
% 36.35/5.05 converse(join(composition(converse(converse(converse(Y))), converse(complement(composition(X, converse(Y))))), converse(complement(X))))
% 36.35/5.05 = { by axiom 1 (converse_idempotence) }
% 36.35/5.05 converse(join(composition(converse(Y), converse(complement(composition(X, converse(Y))))), converse(complement(X))))
% 36.35/5.05 = { by lemma 49 R->L }
% 36.35/5.05 converse(join(composition(converse(Y), complement(converse(composition(X, converse(Y))))), converse(complement(X))))
% 36.35/5.05 = { by lemma 46 R->L }
% 36.35/5.05 converse(converse(join(complement(X), converse(composition(converse(Y), complement(converse(composition(X, converse(Y)))))))))
% 36.35/5.05 = { by axiom 6 (converse_multiplicativity) }
% 36.35/5.05 converse(converse(join(complement(X), composition(converse(complement(converse(composition(X, converse(Y))))), converse(converse(Y))))))
% 36.35/5.05 = { by axiom 3 (maddux1_join_commutativity) R->L }
% 36.35/5.05 converse(converse(join(composition(converse(complement(converse(composition(X, converse(Y))))), converse(converse(Y))), complement(X))))
% 36.35/5.05 = { by axiom 8 (converse_additivity) }
% 36.35/5.05 converse(join(converse(composition(converse(complement(converse(composition(X, converse(Y))))), converse(converse(Y)))), converse(complement(X))))
% 36.35/5.05 = { by lemma 17 }
% 36.35/5.05 converse(join(composition(converse(converse(converse(Y))), complement(converse(composition(X, converse(Y))))), converse(complement(X))))
% 36.35/5.05 = { by axiom 3 (maddux1_join_commutativity) }
% 36.35/5.05 converse(join(converse(complement(X)), composition(converse(converse(converse(Y))), complement(converse(composition(X, converse(Y)))))))
% 36.35/5.05 = { by lemma 49 R->L }
% 36.35/5.05 converse(join(complement(converse(X)), composition(converse(converse(converse(Y))), complement(converse(composition(X, converse(Y)))))))
% 36.35/5.05 = { by axiom 6 (converse_multiplicativity) }
% 36.35/5.05 converse(join(complement(converse(X)), composition(converse(converse(converse(Y))), complement(composition(converse(converse(Y)), converse(X))))))
% 36.35/5.05 = { by lemma 19 }
% 36.35/5.05 converse(complement(converse(X)))
% 36.35/5.05 = { by lemma 49 }
% 36.35/5.05 converse(converse(complement(X)))
% 36.35/5.05 = { by axiom 1 (converse_idempotence) }
% 36.35/5.05 complement(X)
% 36.35/5.05
% 36.35/5.05 Goal 1 (goals_1): join(composition(complement(x2), x1), complement(x0)) = complement(x0).
% 36.35/5.05 Proof:
% 36.35/5.05 join(composition(complement(x2), x1), complement(x0))
% 36.35/5.05 = { by lemma 50 R->L }
% 36.35/5.05 join(composition(complement(x2), x1), join(complement(x0), composition(complement(composition(x0, converse(x1))), x1)))
% 36.35/5.05 = { by axiom 3 (maddux1_join_commutativity) R->L }
% 36.35/5.05 join(composition(complement(x2), x1), join(composition(complement(composition(x0, converse(x1))), x1), complement(x0)))
% 36.35/5.05 = { by axiom 9 (maddux2_join_associativity) }
% 36.35/5.05 join(join(composition(complement(x2), x1), composition(complement(composition(x0, converse(x1))), x1)), complement(x0))
% 36.35/5.05 = { by axiom 12 (composition_distributivity) R->L }
% 36.35/5.05 join(composition(join(complement(x2), complement(composition(x0, converse(x1)))), x1), complement(x0))
% 36.35/5.05 = { by axiom 3 (maddux1_join_commutativity) }
% 36.35/5.05 join(complement(x0), composition(join(complement(x2), complement(composition(x0, converse(x1)))), x1))
% 36.35/5.05 = { by lemma 41 }
% 36.35/5.05 join(complement(x0), composition(complement(meet(x2, composition(x0, converse(x1)))), x1))
% 36.35/5.05 = { by axiom 10 (goals) R->L }
% 36.35/5.05 join(complement(x0), composition(complement(meet(join(composition(x0, converse(x1)), x2), composition(x0, converse(x1)))), x1))
% 36.35/5.05 = { by lemma 41 R->L }
% 36.35/5.05 join(complement(x0), composition(join(complement(join(composition(x0, converse(x1)), x2)), complement(composition(x0, converse(x1)))), x1))
% 36.35/5.05 = { by lemma 40 R->L }
% 36.35/5.05 join(complement(x0), composition(complement(join(zero, meet(join(composition(x0, converse(x1)), x2), composition(x0, converse(x1))))), x1))
% 36.35/5.05 = { by lemma 27 }
% 36.35/5.05 join(complement(x0), composition(complement(join(zero, meet(composition(x0, converse(x1)), join(composition(x0, converse(x1)), x2)))), x1))
% 36.35/5.05 = { by lemma 35 R->L }
% 36.35/5.05 join(complement(x0), composition(complement(join(zero, join(meet(composition(x0, converse(x1)), join(composition(x0, converse(x1)), x2)), zero))), x1))
% 36.35/5.05 = { by lemma 15 R->L }
% 36.35/5.05 join(complement(x0), composition(complement(join(zero, join(meet(composition(x0, converse(x1)), join(composition(x0, converse(x1)), x2)), complement(top)))), x1))
% 36.35/5.05 = { by lemma 23 R->L }
% 36.35/5.05 join(complement(x0), composition(complement(join(zero, join(meet(composition(x0, converse(x1)), join(composition(x0, converse(x1)), x2)), complement(join(join(composition(x0, converse(x1)), x2), top))))), x1))
% 36.35/5.05 = { by axiom 5 (def_top) }
% 36.35/5.05 join(complement(x0), composition(complement(join(zero, join(meet(composition(x0, converse(x1)), join(composition(x0, converse(x1)), x2)), complement(join(join(composition(x0, converse(x1)), x2), join(composition(x0, converse(x1)), complement(composition(x0, converse(x1))))))))), x1))
% 36.35/5.05 = { by axiom 9 (maddux2_join_associativity) }
% 36.35/5.05 join(complement(x0), composition(complement(join(zero, join(meet(composition(x0, converse(x1)), join(composition(x0, converse(x1)), x2)), complement(join(join(join(composition(x0, converse(x1)), x2), composition(x0, converse(x1))), complement(composition(x0, converse(x1)))))))), x1))
% 36.35/5.05 = { by axiom 10 (goals) }
% 36.35/5.05 join(complement(x0), composition(complement(join(zero, join(meet(composition(x0, converse(x1)), join(composition(x0, converse(x1)), x2)), complement(join(join(x2, composition(x0, converse(x1))), complement(composition(x0, converse(x1)))))))), x1))
% 36.35/5.05 = { by axiom 3 (maddux1_join_commutativity) R->L }
% 36.35/5.05 join(complement(x0), composition(complement(join(zero, join(meet(composition(x0, converse(x1)), join(composition(x0, converse(x1)), x2)), complement(join(join(composition(x0, converse(x1)), x2), complement(composition(x0, converse(x1)))))))), x1))
% 36.35/5.05 = { by lemma 39 R->L }
% 36.35/5.05 join(complement(x0), composition(complement(join(zero, join(meet(composition(x0, converse(x1)), join(composition(x0, converse(x1)), x2)), complement(join(meet(top, join(composition(x0, converse(x1)), x2)), complement(composition(x0, converse(x1)))))))), x1))
% 36.35/5.05 = { by lemma 27 }
% 36.35/5.05 join(complement(x0), composition(complement(join(zero, join(meet(composition(x0, converse(x1)), join(composition(x0, converse(x1)), x2)), complement(join(meet(join(composition(x0, converse(x1)), x2), top), complement(composition(x0, converse(x1)))))))), x1))
% 36.35/5.05 = { by axiom 10 (goals) }
% 36.35/5.05 join(complement(x0), composition(complement(join(zero, join(meet(composition(x0, converse(x1)), join(composition(x0, converse(x1)), x2)), complement(join(meet(x2, top), complement(composition(x0, converse(x1)))))))), x1))
% 36.35/5.05 = { by axiom 11 (maddux4_definiton_of_meet) }
% 36.35/5.05 join(complement(x0), composition(complement(join(zero, join(meet(composition(x0, converse(x1)), join(composition(x0, converse(x1)), x2)), complement(join(complement(join(complement(x2), complement(top))), complement(composition(x0, converse(x1)))))))), x1))
% 36.35/5.05 = { by lemma 15 }
% 36.35/5.05 join(complement(x0), composition(complement(join(zero, join(meet(composition(x0, converse(x1)), join(composition(x0, converse(x1)), x2)), complement(join(complement(join(complement(x2), zero)), complement(composition(x0, converse(x1)))))))), x1))
% 36.35/5.05 = { by axiom 3 (maddux1_join_commutativity) }
% 36.35/5.05 join(complement(x0), composition(complement(join(zero, join(meet(composition(x0, converse(x1)), join(composition(x0, converse(x1)), x2)), complement(join(complement(join(zero, complement(x2))), complement(composition(x0, converse(x1)))))))), x1))
% 36.35/5.05 = { by lemma 41 }
% 36.35/5.05 join(complement(x0), composition(complement(join(zero, join(meet(composition(x0, converse(x1)), join(composition(x0, converse(x1)), x2)), complement(complement(meet(join(zero, complement(x2)), composition(x0, converse(x1)))))))), x1))
% 36.35/5.05 = { by lemma 33 }
% 36.35/5.05 join(complement(x0), composition(complement(join(zero, join(meet(composition(x0, converse(x1)), join(composition(x0, converse(x1)), x2)), complement(complement(meet(complement(x2), composition(x0, converse(x1)))))))), x1))
% 36.35/5.05 = { by lemma 27 R->L }
% 36.35/5.05 join(complement(x0), composition(complement(join(zero, join(meet(composition(x0, converse(x1)), join(composition(x0, converse(x1)), x2)), complement(complement(meet(composition(x0, converse(x1)), complement(x2))))))), x1))
% 36.35/5.05 = { by lemma 26 }
% 36.35/5.05 join(complement(x0), composition(complement(join(zero, join(meet(composition(x0, converse(x1)), join(composition(x0, converse(x1)), x2)), meet(meet(composition(x0, converse(x1)), complement(x2)), meet(composition(x0, converse(x1)), complement(x2)))))), x1))
% 36.35/5.05 = { by lemma 43 }
% 36.35/5.05 join(complement(x0), composition(complement(join(zero, join(meet(composition(x0, converse(x1)), join(composition(x0, converse(x1)), x2)), meet(composition(x0, converse(x1)), complement(x2))))), x1))
% 36.35/5.05 = { by axiom 10 (goals) }
% 36.35/5.05 join(complement(x0), composition(complement(join(zero, join(meet(composition(x0, converse(x1)), x2), meet(composition(x0, converse(x1)), complement(x2))))), x1))
% 36.35/5.05 = { by lemma 31 }
% 36.35/5.05 join(complement(x0), composition(complement(join(zero, composition(x0, converse(x1)))), x1))
% 36.35/5.05 = { by lemma 36 }
% 36.35/5.05 join(complement(x0), composition(complement(composition(x0, converse(x1))), x1))
% 36.35/5.05 = { by lemma 50 }
% 36.35/5.05 complement(x0)
% 36.35/5.05 % SZS output end Proof
% 36.35/5.05
% 36.35/5.05 RESULT: Theorem (the conjecture is true).
%------------------------------------------------------------------------------